Topology mediates transport of nanoparticles in macromolecular networks

Diffusion transport of nanoparticles in confined environments of macromolecular networks is common in diverse physical systems and regulates many biological responses. Macromolecular networks possess various topologies, featured by different numbers of degrees and genera. Although the network topologies can be manipulated from a molecular level, how the topology impacts the transport of nanoparticles in macromolecular networks remains unexplored. Here, we develop theoretical approaches combined with simulations to study nanoparticle transport in a model system consisting of network cells with defined topologies. We find that the topology of network cells has a profound effect on the free energy landscape experienced by a nanoparticle in the network cells, exhibiting various scaling laws dictated by the topology. Furthermore, the examination of the impact of cell topology on the detailed behavior of nanoparticle dynamics leads to different dynamical regimes that go beyond the particulars regarding the local network loop. The results might alter the conventional picture of the physical origin of transport in networks.


Supplementary Information
Topology mediates transport of nanoparticles in macromolecular networks

I. Development of analytical model for free energy landscape
To introduce the general formalism of the new analytical approach, we consider a hard particle of radius R in a cross-linked macromolecular network of Gaussian chains without dangling ends. The network topology is specified by (i) the set of cross-links Considering the cross-linked chains in a dilute solution of θ-solvent, where screened excluded volume statistics can be assumed, the interaction potential between monomers is ignored to recover the ideal statistic 1 . Through coupling the excluded effect of the spherical nanoparticle into the renowned theory of network elasticity 2,3 , the partition function of the particle-network system takes the form: where rk is the position vector of the cross-link point k, rij = rij(s) is the path vector of the strand with its start rij(0) = ri and end rij(1) = rj , rnp is the position vector of the nanoparticle, is the delta function, β = 1/kBT, kB is the Boltzmann constant, and T is the temperature. The Hamiltonian of the strand between cross-link pair (i, j) is given by, 2  r r r r r r r r r r , and ij D  r means the path integral over all conformations of the curve rij . In the canonical ensemble, the Helmholtz free energy of the particle-network system is determined by As schemed in Fig. S9, we consider a spherical nanoparticle of radius R at the point O0 (rnp = (x,y,z)), and vectors ri = (xi,yi,zi) and rj = (xj,yj,zj) denote the position of cross-linked points i and j. The spatial position of segments on the curve at s is represented by the collection of threedimensional vectors r = {r1,r2,…,rN}, and the unit vector uij(s) = drij(s)/ds determines the tangent direction. When the chain contacts with the spherical nanoparticle, the partition function of the strand can be rewritten as In this case, we divide the curve into two line segments PT1, QT2 and two arcs T1T0, T0T2 by tangent points T0, T1, and T2. We then obtain the integral from point P to point Q by geometrical relationship, According to the geometric relationship, the lengths are given in the following, Hence, the solution of Eq. S12 can be written as  where m = i or j. Therefore, Eq. S13 can be transformed into

II. Minimal energy path of the transition of a nanoparticle between two neighboring cells
In this section, we provide a detailed evidence regarding that the linear connection between the core region of a network cell and the local free energy minimum at the center of a cell face, which is defined as z axis in the main text, is one of the minimum energy paths (MEPs).
As schemed in Fig. S9, we still consider a spherical nanoparticle of radius R at the point where the rotation matrix k n C denotes rotations of 2/n, performed k times. Using the rotation matrix, we can extract a group of chains, p, with the positions of the chain ends {rik}, {rjk} given r . Using the Eq. S4, the free energy of the total chains in group p is given Now we prove that for any point along z axis, , the profiles of free energy ( ) np F r satisfy the saddle point condition, i.e., the differential of free energy is a small displacement perpendicular to z axis. For this purpose, we consider that the small displacement np r is given to a nanoparticle at a position along z axis, the new position can be written as np np np    r r r .
Using Eq. S16, we get By substituting Eq. S18 into Eq. S17, the differential of the free energy of chains can be calculated as ij i j np L r r r represents the minimal value of the path integral of the curve ij r , and    r r r r r r r r r r is its differential.
In combination with Eqs. S10, S13 and S15, the minimal value of the integral ( , , ) ij i j np L r r r is obtained by setting the dihedral angle =0, taking the form (1) For R l  , the differential of Eq. S20 is given by (2) For f l R R   , the differential of Eq. S20 takes the form: where the vectors M1, M2, M3 are r r r r r r r M r r r r r r r r By substituting Eq. S18 to Eq. S22, the differential of free energy is given by (3) For f R R  , the differential of Eq. S20 can be written as: r r r r r r r r r r r r r r r r r (S25) Substituting Eq. S18 into Eq. S25, the differential is given by r r r r r r r r r C r C r r r r r r r (S26) Clearly, keeps in all regimes. Hence, the differential of free energy can be expressed as This clarifies that any point on z-axis is saddle point, corroborating that z-axis is one of the MEPs.

III. Development of analytical model for microscopic dynamics in different regimes
The microscopic dynamics of the nanoparticle is theoretically well described by a nonlinear Langevin equation 6  In order to extract the dynamics described by Eq. S28 in the periodic potentials of Regimes I, II and III, we need to coarse-grain and decompose it into a series of consecutive jump and waiting events, where the nanoparticle spends most of the time close to a cell center and only occasionally escapes to another one. We identify these trapping periods with the waiting times and the escape events with the jumps of the continuous time random walk (CTRW) 7 . It has to wait for a random waiting time drawn from the probability distribution function (PDF) of the waiting time (t), before it makes a jump, and the length of the jump can be chosen to be a random variable, δz, distributed in terms of the PDF, λ(δz). For the present discussion, we assume that the CTRW process is separable, in the sense that t and δz are independently identically distribution (IID). The position of the nanoparticle is given by, where N(t) is the counts of transition events at time t, and δzi is the jump length of the event i.
The distribution of the total number of network cells traversed can accordingly be obtained from the Montroll-Weiss equation 7   In order to accurately predict the trajectory of the nanoparticle, a waiting time distribution (t) can be obtained in exponential distribution via Poisson processes Combining Eqs. S32 and S38 and performing inverse Laplace transformation, the average number of jumps gives ( ) / w N t t t  . Therefore, Eq. S37 can be simplified as The characteristic waiting time w t can be determined by Kramers' rate theory 8 , where ks and kl are the curvatures of the potential along z-axis at the minimum point z = 0 and the maximum point z = rin, respectively.
In view of Eq. S39, for the long time scale, MSD recovers back to the normal diffusion.
For the short time scale with t < tw, , indicating that the nanoparticle displacement is equivalent to the displacement of a nanoparticle in a single cell, where hopping events don't happen and it is unnecessary to distinguish between z  and z. In the latter case, the microscopic dynamics of the nanoparticle in different regimes will significantly depend on the form of the energy landscape, which is discussed as follow: Regime I: The potential F(z) = ρ|z| and is "V-shaped", with ρ=Ub/rin being an arbitrary constant 9 . Thus, for a nanoparticle diffusing in a potential F(z) the Fokker-Planck equation describing the dynamics of the PDF P(z, t) under a constant resetting rate r=1/tw reads where 0 B D = k T/6 R   represents the diffusion coefficient of the nanoparticle in solvents.
Taking the Laplace transformation of Eq. S41, The solution to the above equation reads At t=0 + , the starting condition of Eq. S42 can be written as From the solution Eq. S48, we can calculate the MSD For the symmetric case (z0 = 0), Eq. S49 reduces to the expression Given that the characteristic equation With a constant particular solution 2 2 ( ) / B z t k T m  , the complete solution of Eq. S54 can be written as Through the initial conditions 2 ( ) 0 z t  , 2 ( ) 0 d z t dt  at t=0, the exact values of constants C1 and C2 in Eq. S57 are determined as Such that the Eq. S57 can be transformed into (b) In the underdamped case ( 8m    ), the solution reads We set the initial condition z(0) = 0, and thus the MSD in Regime II response, i.e., |v|>0, ||>0, the oscillation mode changes into underdamped; if a system is undamped with oscillatory response, i.e., |v|=0, |>0, the oscillation mode becomes harmonic.
However, if the inertia effects are negligible by setting 2 The solution of which takes the purely exponential form, e −vt , resulting in the overdamped mode of oscillation as prevailed in Brownian dynamics.
As discussed in the above section, the diffusion dynamics of Regimes I, II and III exhibit various oscillation modes, highly correlating to the forms of the free energy landscape F(z) in In Regime II, affected by partial chains of the cell, the nanoparticle turns to receiving negative response with a linear restoring forcez, which competes with the frictional force. When the restoring force is large, this corresponds to obvious oscillatory behavior. By contrast, when the restoring force becomes small, nanoparticle dynamics is essentially diffusive. In Regime III where the nanoparticle experiences the free energy at the boundary of the cell, the response reduces to zero. Therefore, the dynamics in Regime III is dominated only by the frictional force, and the oscillation is overdamped.
However, as only a local network loop is considered in previous works, the viscosity force of the nanoparticle is far larger than the restoring force around z = 0. Thus, oscillatory behavior of the nanoparticle is suppressed and the underdamped modes of oscillation cannot be observed.

VI. Monte Carlo (MC) simulation
For a macromolecular network with polyhedral cells, the standard coordinates ri and the linker connection rij can be found in ref. 5